#388485

order	512
nilpotency class	3
isoclinism	isoclinic class of #30871
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,11,17)(7,18,23,32)(8,9,14,29)(10,16,30,15)(12,22,33,21)(13,25,28,26)(19,27,35,31)(20,24,36,34);(5,7)(6,12)(8,24)(9,27)(10,22)(11,23)(13,20)(14,34)(15,32)(16,18)(17,33)(19,26)(21,30)(25,35)(28,36)(29,31);(1,3,4,2)(5,8)(6,13)(7,19)(9,16)(10,25)(11,14)(12,31)(15,29)(17,28)(18,20)(21,34)(22,24)(23,35)(26,30)(27,33)(32,36);(5,9,10,13,11,29,30,28)(6,14,15,26,17,8,16,25)(7,20,21,27,23,36,22,31)(12,19,18,34,33,35,32,24)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295681);